Properties

Label 2-115920-1.1-c1-0-84
Degree $2$
Conductor $115920$
Sign $-1$
Analytic cond. $925.625$
Root an. cond. $30.4240$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s − 2·11-s − 2·13-s − 4·17-s + 2·19-s − 23-s + 25-s + 2·31-s − 35-s + 10·37-s − 2·41-s − 6·43-s − 2·47-s + 49-s − 6·53-s − 2·55-s + 4·59-s + 14·61-s − 2·65-s + 10·67-s − 2·71-s − 6·73-s + 2·77-s + 16·79-s − 6·83-s − 4·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.603·11-s − 0.554·13-s − 0.970·17-s + 0.458·19-s − 0.208·23-s + 1/5·25-s + 0.359·31-s − 0.169·35-s + 1.64·37-s − 0.312·41-s − 0.914·43-s − 0.291·47-s + 1/7·49-s − 0.824·53-s − 0.269·55-s + 0.520·59-s + 1.79·61-s − 0.248·65-s + 1.22·67-s − 0.237·71-s − 0.702·73-s + 0.227·77-s + 1.80·79-s − 0.658·83-s − 0.433·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115920\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(925.625\)
Root analytic conductor: \(30.4240\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{115920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 + T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.79306182964298, −13.30958983080000, −12.87611920051396, −12.61533644627254, −11.82053399653081, −11.41370259181724, −11.01309317203354, −10.24793727243865, −9.976881571019166, −9.534670192763309, −9.005357176641486, −8.400417668328143, −7.924871390136410, −7.412601147556734, −6.659677390182760, −6.506222807041395, −5.760847710818430, −5.189122488642319, −4.803674118932069, −4.103173256557259, −3.510369587515474, −2.674807285515642, −2.451425816771574, −1.667226867337641, −0.7964394788630319, 0, 0.7964394788630319, 1.667226867337641, 2.451425816771574, 2.674807285515642, 3.510369587515474, 4.103173256557259, 4.803674118932069, 5.189122488642319, 5.760847710818430, 6.506222807041395, 6.659677390182760, 7.412601147556734, 7.924871390136410, 8.400417668328143, 9.005357176641486, 9.534670192763309, 9.976881571019166, 10.24793727243865, 11.01309317203354, 11.41370259181724, 11.82053399653081, 12.61533644627254, 12.87611920051396, 13.30958983080000, 13.79306182964298

Graph of the $Z$-function along the critical line